Advertisements
Advertisements
प्रश्न
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Advertisements
उत्तर
Let R be the radius and h be the height of the cylinder which is inscribed in a sphere of radius r cm.
Then from the figure,
`"R"^2 + (h/2)^2` = r2
∴ R2 = `r^2 - h^2/(4)` ...(1)
Let V be the volume of the cylinder.
Then V = πR2h
= `pi(r^2 - h^2/(4))h` ...[By (1)]
= `pi(r^2 - h^3/(4))`
∴ `"dV"/"dh" = pid/"dh"(r^2h - h^3/(4))`
= `pi(r^2 xx 1 - 1/4 xx 3h^2)`
= `pi(r^2 - 3/4h^2)`
and
`(d^2V)/("dh"^2) = pid/"dh"(r^2 - 3/4h^2)`
= `pi(0 - 3/4 xx 2h)`
= `-(3)/(2)pih`
Now, `"dV"/"dh" = 0 "gives", pi(r^2 - 3/4h^2)` = 0
∴ `r^2 - 3/4h^2` = 0
∴ `(3)/(4)h^2` = r2
∴ h2 = `(4r^2)/(3)`
∴ h = `(2r)/sqrt(3)` ...[∵ h > 0]
and
`((d^2V)/(dh^2))_("at" h = (2r)/sqrt(3)`
= `-(3)/(2)pi xx (2r)/sqrt(3) < 0`
∴ V is maximum at h = `(2r)/sqrt(3)`
If h = `(2r)/sqrt(3)`, then from (1)
R2 = `r^2 - (1)/(4) xx (4r^2)/(3) = (2r^2)/(3)`
∴ volumeof the largest cylinder
= `pi xx (2r^2)/(3) xx (2r)/sqrt(3) = (4pir^3)/(3sqrt(3)`cu cm.
Hence, the volume of the largest cylinder inscribed in a sphere of radius 'r' cm = `(4pir^3)/(3sqrt(3)`cu cm.
APPEARS IN
संबंधित प्रश्न
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x2
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
g(x) = x3 − 3x
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = x/2 + 2/x, x > 0`
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Solve the following : Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/(3)`.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The function f(x) = x log x is minimum at x = ______.
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
Range of projectile will be maximum when angle of projectile is
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
The rectangle has area of 50 cm2. Complete the following activity to find its dimensions for least perimeter.
Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
Now,f'(x) = 0, if x = `square`
But x is not negative.
∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`
∴ by the second derivative test f is minimum at x = `root(5)(2)`
When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`
∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
